De Sitter Universe - Mathematical Expression

Mathematical Expression

A de Sitter universe has no ordinary matter content but with a positive cosmological constant which sets the expansion rate, . A larger cosmological constant leads to a larger expansion rate:

where the constants of proportionality depend on conventions. The cosmological constant is .

It is common to describe a patch of this solution as an expanding universe of the FLRW form where the scale factor is given by

where the constant is the Hubble expansion rate and is time. As in all FLRW spaces, the scale factor, describes the expansion of physical spatial distances.

Unique to universes described by the FLRW metric, a de Sitter universe has a Hubble Law which is not only consistent through all space, but also through all time (since the deceleration parameter is equal to ), thus satisfying the perfect cosmological principle that assumes isotropy and homogeneity throughout space and time. As a class of models with different values of the Hubble constant, the static universe that Einstein developed, and for which he invented the cosmological constant, can be considered a special case of the de Sitter universe where the expansion is finely tuned to just cancel out the collapse associated with the positive curvature associated with a non-zero matter density. There are ways to cast de Sitter space with static coordinates (see de Sitter space), so unlike other FLRW models, de Sitter space can be thought of as a static solution to Einstein's equations even though the geodesics followed by observers necessarily diverge in the normal way expected from the expansion of physical spatial dimensions. As a model for the universe, de Sitter's solution was not considered viable for the observed universe until models for inflation and dark energy were developed. Before then, it was assumed that the Big Bang implied only an acceptance of the weaker cosmological principle which holds isotropy true only for spatial extents but not temporal extents.